Journal of Homotopy and Related Structures最新文献

We consider moment-angle complexes associated with skeleta of simplices and determine the homotopy type of their quotient spaces under the diagonal circle action.

For any object A in a simplicial model category (mathcal {M}), we construct a topological space (hat{A}) which classifies homogeneous functors whose value on k open balls is equivalent to A. This extends a classification result of Weiss for homogeneous functors into topological spaces.

A Loday–Pirashvili module over a Lie algebra (mathfrak {g}) is a Lie algebra object (bigl (Gxrightarrow {X} mathfrak {g}bigr )) in the category of linear maps, or equivalently, a (mathfrak {g})-module G which admits a (mathfrak {g})-equivariant linear map (X:Grightarrow mathfrak {g}). We study dg Loday–Pirashvili modules over Lie algebras, which is a generalization of Loday–Pirashvili modules in a natural way, and establish several equivalent characterizations of dg Loday–Pirashvili modules. To provide a concise characterization, a dg Loday–Pirashvili module is a non-negative and bounded dg (mathfrak {g})-module V paired with a weak morphism of dg (mathfrak {g})-modules (alpha :Vrightsquigarrow mathfrak {g}). Such a dg Loday–Pirashvili module resolves an arbitrarily specified classical Loday–Pirashvili module in the sense that it exists and is unique (up to homotopy). Dg Loday–Pirashvili modules can also be characterized through dg derivations. This perspective allows the calculation of the corresponding twisted Atiyah classes. By leveraging the Kapranov functor on the dg derivation arising from a dg Loday–Pirashvili module ((V,alpha )), a (hbox {Leibniz}_infty [1]) algebra structure can be derived on (wedge ^bullet mathfrak {g}^vee otimes V[1]). The binary bracket of this structure corresponds to the twisted Atiyah cocycle. To exemplify these intricate algebraic structures through specific cases, we utilize this machinery to a particular type of dg Loday–Pirashvili modules stemming from Lie algebra pairs.

A space X is “sequentially n-connected” at (xin X) if for every (0leqslant kleqslant n) and sequence of k-loops (f_1,f_2,f_3,ldots :S^krightarrow X) that converges toward the point x, the maps (f_m) contract by a sequence of null-homotopies that converge toward x. Unlike standard local contractibility conditions, the sequential n-connectedness property is closed under forming infinite products and infinite shrinking wedges. We use this property, in conjunction with the Whitney Covering Lemma, to construct homotopies that simultaneously perform infinite deformations of n-loops and, ultimately, allow us to continuously deform arbitrary n-loops into maps with simpler forms. As a direct application, we extend the computation of the n-th homotopy group of a shrinking wedge of certain ((n-1))-connected spaces due to K. Eda and K. Kawamura.

By a theorem of Christensen and Hovey, the category of non-negatively graded chain complexes has a model structure, called the h-model structure or Hurewicz model structure, where the weak equivalences are the chain homotopy equivalences. The Dold–Kan correspondence induces a model structure on the category of simplicial modules. In this paper, we give a description of the two model categories and some of their properties, notably the fact that both are monoidal.

In this paper, we study the (mathbb {Z}/2) action on complex Grassmann manifolds (G_{n}(mathbb {C}^{2n})) given by taking orthogonal complement. We completely compute the associated (mathbb {Z}/2) Fadell–Husseini index. Our study is parallel to the study of the index of real Grassmann manifolds (G_n(mathbb {R}^{2n})) by Baralić et al. [Forum Math., 30 (2018), pp. 1539–1572].

This paper is a follow-up to Positselski and Št’ovíček (Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity. Electronic preprint arXiv:2212.09639 [math.AG]). We consider two algebraic settings of comodules over a coring and contramodules over a topological ring with a countable base of two-sided ideals. These correspond to two (noncommutative) algebraic geometry settings of certain kind of stacks and ind-affine ind-schemes. In the context of a coring ({mathcal {C}}) over a noncommutative ring A, we show that all A-flat ({mathcal {C}})-comodules are (aleph _1)-directed colimits of A-countably presentable A-flat ({mathcal {C}})-comodules. In the context of a complete, separated topological ring ({mathfrak {R}}) with a countable base of neighborhoods of zero consisting of two-sided ideals, we prove that all flat ({mathfrak {R}})-contramodules are (aleph _1)-directed colimits of countably presentable flat ({mathfrak {R}})-contramodules. We also describe arbitrary complexes, short exact sequences, and pure acyclic complexes of A-flat ({mathcal {C}})-comodules and flat ({mathfrak {R}})-contramodules as (aleph _1)-directed colimits of similar complexes of countably presentable objects. The arguments are based on a very general category-theoretic technique going back to an unpublished 1977 preprint of Ulmer and rediscovered in Positselski (Notes on limits of accessible categories. Electronic preprint arXiv:2310.16773 [math.CT]). Applications to cotorsion periodicity and coderived categories of flat objects in the respective settings are discussed. In particular, in any acyclic complex of cotorsion ({mathfrak {R}})-contramodules, all the contramodules of cocycles are cotorsion.

We give a very simple construction of the string 2-group as a strict Fréchet Lie 2-group. The corresponding crossed module is defined using the conjugation action of the loop group on its central extension, which drastically simplifies several constructions previously given in the literature. More generally, we construct strict 2-group extensions for a Lie group from a central extension of its based loop group, under the assumption that this central extension is disjoint commutative. We show in particular that this condition is automatic in the case that the Lie group is semisimple and simply connected.

With the goal of transferring dg algebra structures on complexes along contractions, we introduce a new condition on the associated homotopy, namely a generalized version of the Leibniz rule. We prove that, with this condition, the transfer works to yield a dg algebra (with vanishing descended higher (A_infty ) products) and prove that it works also after an application of the Perturbation Lemma even though the new homotopy may no longer satisfy that condition. We also extend these results to the setting of (A_infty ) algebras. Then we return to our original motivation from commutative algebra. We apply these methods to find a new method for building a dg algebra structure on a well-known resolution, obtaining one that is both concrete and permutation invariant. The naturality of the construction enables us to find dg algebra homomorphisms between these as well, enabling them to be used as inputs for constructing bar resolutions.

We classify all 2-term (L_infty )-algebras up to isomorphism. We show that such (L_infty )-algebras are classified by a Lie algebra, a vector space, a representation (all up to isomorphism) and a cohomology class of the corresponding Lie algebra cohomology.